March 30, 2017
Ronan Herry,
Optimal transport and spaces with Ricci curvature bounded below
Abstract:
Given a Riemannian manifold (that is a particular metric space on which the notion volume can be defined) a very natural question is to understand how the volume of balls varies depending on its radius and/or its center.
Formalizing this idea of the interplay between distances and volumes led to a rich theory known as "theory of Riemannian manifolds with Ricci curvature bounded below". Rather than focusing on technical details, I will illustrate the power of this theory with numerous examples.
On the other hand one can see a volume element as a distribution of mass that one wants to transport (reorganize) to an other volume element (distribution of mass). Given a cost of moving volume elements, one might wonder what is the most cost-effective way of transporting the mass.
This problem is formally known as "optimal transport". When the cost depends explicitly on the distance, optimal transport strongly intertwines volumes and distances.
The volume-distance intrication appearing both in optimal transport and the theory of Riemannian manifolds with Ricci curvature bounded below suggests a strong link between the two theories. In the last 15 years, this link has been discovered and explicated : any statement of the theory of Ricci curvature bounded below can be rephrased in an optimal transport fashion.
The optimal transport point of view does not require any smoothness or regularity assumptions and can therefore be used to study rougher spaces.
April 6, 2017
Emiliano Torti,
About the mathematical legacy of S. Ramanujan: Congruences between modular forms.
Abstract:
In his article “On certain arithmetical function” in 1916, S. Ramanujan defines the arithmetic function Tau. First, we will present his results concerning the arithmetic properties of this function, and later we will restate them in terms of the modern language of modular forms. Secondly, we will discuss open questions that arose at the end of his work . Ramanujan’s achievements inspired the work of two Fields medalist J.P. Serre and P. Deligne who were the first ones to clarify the deep nature of his results. Their work gave birth to the theory of Galois representations attached to modular forms which is now a central topic in number theory and for instance, it is the main tool in A. Wiles proof of Fermat’s Last Theorem.
April 13, 2017
Assar Andersson,
Operads of compactified configuration spaces.
Abstract:
In this talk we are going to discuss how different types of algebras can be encoded by an algebraic object called operad. Then we are going to show how a certain compactification of configuration spaces can give us a geometric interpretation of homotopy (Lie and associative)-algebras. This interpretation can also serve as an independent description of what a homotopy Lie-algebra is
(so the audience does not have to know that beforehand). We will also outline how we can use this interpretation to construct interesting algebraic structures.
April 27, 2017
Filippo Mazzoli,
A parametrization of complex projective structures on surfaces.
Abstract: The aim of this talk will be to give an introduction of the concept of complex projective structures on surfaces and to describe a parametrization result of them. We will firstly give the necessary definitions and we will describe how to build the developing map and the holonomy representation of such a structure. Finally, we will discuss the connection between these objects and the data of a Riemann surface structure and a holomorphic quadratic differential on it.
May 4, 2017
Robert Baumgarth ,
Stochastic Flow Processes and Brownian Motion on Manifolds.
Abstract: We study stochastic flow processes and diffusions on smooth Riemannian manifolds starting from the well-known notion of a flow to a vector field. As an application we sketch how these concepts can be used to give a very simple proof for existence and uniqueness of a solution to the Dirichlet problem.
We give a brief overview how to define Brownian motion on (smooth Riemannian) manifolds: the extrinsic approach as solution to the usual martingale problem using a Whitney embedding and the Eells-Elworthy-Malliavin approach using the projection from the orthonormal frame bundle.
All notions will be briefly introduced during the talk as needed concerning the broad audience.
May 11, 2017
Jimmy Devillet,
Characterizations of idempotent uninorms.
Abstract: Aggregation functions defined on linguistic scales (i.e., finite chains) have been intensively investigated for about two decades. Among these functions, fuzzy connectives (such as uninorms) are binary operations that play in an important role in fuzzy logic.
In this talk we focus on the characterization of the class of idempotent uninorms on finite chains. Indeed, we provide an axiomatic characterization of the idempotent uninorms in terms of three conditions only : quasitriviality, symmetry and non-decreasing monotonicity. Moreover, we provide a graphical characterization of these operations in terms of their contour plots. Finally, we present an algebraic translation of the previous graphical characterization in terms of single-peaked linear orderings.
May 18, 2017
Damjan Pistalo ,
Model category theory and applications.
Abstract: In generality, homotopy theory is the study of mathematical contexts in which functions are equipped with a concept of homotopy between them. A key aspect of the theory is that the concept of isomorphism is relaxed to that of homotopy equivalence: Where a function is regarded as invertible if there is a reverse function such that both composites are equal to the identity, for a homotopy equivalence one only requires the composites to be homotopic to the identity. Model categories give rise to a large class of homotpy theories, and provide a setting that is suitable for concrete calculations.
My aim is to explain the basics of category theory, define model categories and show how to work in this setting. I will state the Quillen transfer theorem and illustrate it in the theory of homological algebra. Given the time, I will say something about the derived critical locus in the derived algebraic geometry.
This semester the seminar will focus of Free probability following the book of Nica & Speicher.
September 15, 2016
Ronan Herry & François Petit,
Organisational meeting
September 29, 2016
Ronan Herry,
Intuition for free probability, building by hands non-commutative probability spaces, examples, the spectral theorem (Tao 1 & Speicher 1)
October 6, 2016
Guangqu Zheng,
Formal definitions of a *-algebra, normal and non-normal distributions and the extended example of chap 2 (Speicher 2)
October 13, 2016
François Petit,
C* algebra: Functional calculus in C*-algebra C*-probability spaces (Speicher 3)
October 20, 2016
Assar Andersson,
Distributions (I): *-distribution, norm, spectrum; Joint distributions (Speicher 3 & 4)
October 27, 2016
Anna Vidotto,
Distributions (II): Joint *-distributions; Joint *-distributions and isomorphisms
November 3, 2016
Olivier Elchinger,
Free independence (I): Tensor independence; Definition; Free product of groups (Speicher 5)
November 10, 2016
Olivier Elchinger,
Free independence (II): Free independence and joint moments; Basic properties; Other universal product (Speicher 5)
November 17, 2016
Maurizia Rossi,
Free product (I): Free product of unit algebra and of non-commutative probability spaces (Speicher 6)
November 24, 2016
Gergely Kiss,
Free product (II): Free product of *-probability spaces (Speicher 6)
December 1, 2016
Yannick Voglaire,
Free product (III): GNS representation and free product of C*-probability spaces (Speicher 7)
December 8, 2016
Christian Döbler,
Free central limit theorem (Speicher 8)
December 15, 2016
Guangqu Zheng,
Semicircular systems and full Fock space (Speicher 7)